# An enlarged six-row calibration region

This establishes Conjecture S from the [concept note](../../concepts_2026_09_10/CONCEPTS.md)
for the same experiment. No new measurements or changed kernel coefficients are used.

## 1. Fixed model and region

Let both reflecting path Laplacians have 1,001 vertices, let
`x_i=(i+1/2)/1001`, and define

\[
 H(g)=I_y\otimes L_x+L_y\otimes\operatorname{diag}(1+gx_i),
 \qquad
 A_j(t,g)=(1+t)^{1/4}R(I_y\otimes e^T)e^{-tH(g)}(\delta_j\otimes U).
\]

The vector `e` is the unit constant vector, and `U` has the two orthonormal
midpoint cosine columns with mode numbers one and two. The selected physical
rows are **485, 493, 499, 501, 507, 515**. Every target `j=490,...,510` and
every nonzero real source `u in R^2` are admitted. Sensor noise has Euclidean
norm at most `eta ||u||`, where `eta=3e-8`. These are spatial-average rows.

Nominal time `t` and true time `t+dt` both lie in `[1,2]`. The potential is
`g=4/5+dg`. Fix units `z1=dt/1e-8`, `z2=dg/1e-8`, and let

\[
 Z=[-2,2]^2,\qquad h=d=2\cdot10^{-8}.
\]

This region is compact, centrally symmetric and convex, contains the original
`[-1,1]^2`, and has area 16. The endpoint restriction on true time remains
part of the model. The area comparison uses the declared clock/potential
coordinates; it is not invariant under a change of units.

## 2. Tightening the complete nominal approximation budget

Let `P_j(t)` be the exact rational midpoint degree-32 kernel from the earlier
reconstruction, and `B_j(t)=alpha(t)P_j(t)`, where `alpha=(1+t)^(1/4)`.
All coefficients are unchanged.

The [original complete model proof](../../transfer_theorem_2026_09/resolution/PROOFS.md)
gives, for every unnormalized matrix entry,

\[
 |A_j/\alpha-P_j|\le e_0=2^{-33}+E_{\rm image}+E_{\rm boundary}+2\cdot10^{-30}.
\]

This is a complete remainder: `2^-33` is the omitted time series on all of
`[1,2]`; the coefficient midpoint error sums every saved coefficient error;
and the other terms cover all omitted periodic images and boundary walks.
Explicitly, with `a=8` and `m=56/5`,

\[
 E_{\rm image}\le
 e^{(18/5)(a+a^{-1}-2)}\frac{a^{-38}+a^{-90}}{1-a^{-64}}<10^{-24},
 \qquad
 E_{\rm boundary}\le\frac{2m^{491}}{491!(1-m/492)}<10^{-500}.
\]

For clarity, the time remainder follows from Taylor's integral formula and
`||H^33 e^{-sH}||/33! <= 1` for positive semidefinite `H` and `s>=1`.
The scalar inequality follows from `exp(lambda)>=lambda^33/33!`.
The coefficient reconstruction encloses the exact finite-x exponential
before this time truncation. The fresh 320-bit replay checks all 1,782
saved coefficient intervals against that construction.

There are **12 entries** in this six-row, two-source matrix. Since
`alpha<4/3`, the same error theorem therefore gives the stronger usable budget

\[
 \|A_j(t,4/5)-B_j(t)\|^2
 \le\frac{16}{9}\,12e_0^2
 <\left(6\cdot10^{-10}\right)^2=:\rho^2.
\]

The previous consumer rounded this allowance to `1e-9`. The new consumer
checks the displayed stricter inequality with exact rational arithmetic.
It evaluates the exponential upper bound by a 200-term positive rational
Taylor sum with the complete geometric remainder. This tightening changes
the certificate, not the approximation or experiment.

## 3. Joint timing and potential incidence

The [complete derivative proof](../../structured_transfer_2026_09/resolution/PROOFS.md)
and its reconstructed degree-32 potential derivative kernel `Q_j` supply

\[
 \|A_{j,g}-\alpha Q_j\|<10^{-12},\qquad
 \|A_{j,t}-B'_j\|<10^{-12}.
\]

These derivative statements come from differentiated reconstruction and
complete image, boundary and Cauchy tails. They do not follow by
differentiating a uniform value-error inequality. The preserved consumer
rechecks all these remainders; a fresh 384-bit differentiated-kernel replay
also reconstructs every saved coefficient interval.

On the enlarged rectangle, the potential remains positive and
`||H(g)|| <= 56/5+4d < 45/4`. Thus the previously proved derivative bounds
remain applicable throughout the entire new family:
`||A_tt||<1`, `||A_tg||<128`, and the potential second-order remainder
coefficient is `128/3`. A sequential Taylor expansion gives

\[
 A_j(t+dt,4/5+dg)=B_j+dt B'_j+dg\,\alpha Q_j+E_j,
 \qquad \|E_j\|\le\rho+r,
\]
\[
 r=h^2/2+128hd+(128/3)d^2+(h+d)10^{-12}
 <6.846671\cdot10^{-14}.
\]

The mixed incidence is charged once. Enlarging both coordinates by two
multiplies the nonlinear second-order terms by four and the derivative
approximation term by two.

Write `D_j=dt B'_j+dg alpha Q_j`. The preserved six-row certificate proves,
over all targets and all times, on the old unit box,

\[
 \sup\|D_j\|<4.3\cdot10^{-10},\qquad
 \sup\|B_j^\dagger D_j\|<6.8\cdot10^{-7}.
\]

The new consumer reruns all **2,688** outward time-cell calculations and
checks every recorded bound against independently translated polynomial
expansions. It also checks every kernel binding. Linearity in the **joint**
parameter vector then gives, exactly on `Z=2[-1,1]^2`,

\[
 F<8.6\cdot10^{-10},\qquad V<1.36\cdot10^{-6}.
\]

This scaling retains the shared clock/potential incidence. It does not
assume that a direction weak at one target is weak at every target.

## 4. Target identification and source recovery

The existing polynomial source floor is

\[
 B_j^TB_j\succ\mu I,\qquad \mu=361/(256\cdot10^9).
\]

At every time, every unequal target pair has a rational split `0<a<1` such
that, writing `alpha0=19/16` and `delta*=3.25e-8`,

\[
 [P_j,-P_k]^T[P_j,-P_k]\succ
 \operatorname{diag}\left(\frac{(\delta_*/\alpha_0)^2}{a}I_2,
 \frac{(\delta_*/\alpha_0)^2}{1-a}I_2\right).
\]

All 1,280 records covering the 21 source cases and 210 pair cases are
rechecked by exact fraction-free determinant elimination and interval
polynomial evaluation. This is the same pair tube; no larger unproved pair
margin is substituted.

Keep the verified normalization error `xi<=1e-9`. The full physical nominal
tube radius satisfies

\[
 \delta=\eta+\rho+\xi+r+F
 <3.246006847\cdot10^{-8}<\delta_*.
\]

If two explanations with unequal targets produced the same data, their
nominal difference would have norm at most
`delta (||u||+||v||)`. The displayed weighted Gram inequality contradicts
this by `(||u||+||v||)^2 <= ||u||^2/a+||v||^2/(1-a)`.
Each rival explanation may have **its own** clock and potential value in
the admitted region. Joint incidence is required within each explanation.

Once the label is determined, nominal least squares gives

\[
 \frac{\|\widehat u-u\|}{\|u\|}
 \le V+\frac{\eta+\rho+\xi+r}{\sqrt\mu}
 <0.000842860553<10^{-3}.
\]

All scalar gates and outward square roots are rationally checked. This
proves Conjecture S, with fourfold rather than merely doubled area.
The coarse source bound `delta/sqrt(mu)<0.000864402100` also passes; the
directional inverse improves the bound but is not needed for this contract.

The decoder accepts exact rational raw readings. It verifies the existing
quartic normalization receipt at **nominal** time, charges `xi`, and applies
the nominal relative-tube feasibility test. Positive semidefiniteness gives
the raw gain bound `4/3+eta` even at uncertain true time, so that wrapper
remains valid. Source estimates use ordinary nominal least squares, while
tube feasibility uses the distinct completed-quadratic solve. The supplied
clock/potential intervals are premises; the decoder does not infer them.

## 5. A quantified calibration frontier, with a limited interpretation

For homothetic squares `Z_s=[-s,s]^2`, `0<=s<=3`, write

\[
 r(s)=s^2\bigl(\tfrac12+128+\tfrac{128}{3}\bigr)10^{-16}
       +2s\cdot10^{-20}.
\]

The current sufficient label gate is the scalar inequality

\[
 3\cdot10^{-8}+6\cdot10^{-10}+10^{-9}
 +4.3s\cdot10^{-10}+r(s)<3.25\cdot10^{-8}.
\]

The same source gate uses `6.8s*1e-7` as its directional term. Exact checks
give:

| Scale | Normalized area | Label tube upper bound | Source upper error | Result |
|---|---:|---:|---:|---|
| 2 | 16 | <3.246006847e-8 | <0.000842860553 | Both pass |
| 2.09 | 17.4724 | <3.249877477e-8 | <0.000842921920 | Both pass |
| 2.10 | 17.64 | >3.250307548e-8 | <0.000842928740 | This label bound fails |

The left side increases strictly with nonnegative `s`, so its unique
crossing lies between 2.09 and 2.10. The fixed raw decoder exposes scale 2;
the scale-2.09 row is an additional analytic consequence, not its input
contract. No actual ambiguity witness is supplied at scale 2.10. The table
locates the frontier of **these sufficient inequalities**, not the intrinsic
limit of the six-row experiment and not an optimal uncertainty shape.

With the former rounded `rho=1e-9`, even scale 2 would fail this label
inequality. This makes the improvement a concrete example of reducing
certificate conservatism while retaining the same experiment. The region
is a mathematical calibration contract, not a protocol for obtaining
the stated joint uncertainty in a physical apparatus.